# Innocuous angular integal

1. Dec 2, 2009

### sv3t

Hi,
Can anyone help me to do the following integral over the unit sphere:
$$\nonumber \int d^2 n \exp\left[-(\vec{a}\cdot \hat n)(\vec{b}\cdot \hat n)-i \,\hat n\cdot \vec{c}\right]$$

where $$\hat n$$ is a unit 3-vector, and $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ are arbitrary real valued constant 3-vectors of arbitrary length. To avoid any unnecessary confusion, $$i\equiv \sqrt{-1}$$; and $$d^2 n=d\Omega=\sin(\theta)d\theta d\phi$$ is the solid angle measure.

Any type of solution/suggestion for a solution (which is hopefully numerically stable) would be greatly appreciated. That includes any type of expansion applied to the integrand; converting it to a PDE; using complex calculus tricks, etc. etc. Of course, an analytical solution would be best.

I lost a huge amount of time on this, but without much success. The only expansion scheme I applied successfully, works for large $$|\vec c|$$; but making any trick work for large $$|\vec a| |\vec b|$$ (which produces a positive quadratic term in the exponent) so far has been impossible for me.

Thanks!